In this work, we explore the limiting dynamics of deep neural networks trained with stochastic gradient descent (SGD). As observed previously, long after performance has converged, networks continue to move through parameter space by a process of anomalous diffusion in which distance traveled grows as a power law in the number of gradient updates with a nontrivial exponent. We reveal an intricate interaction among the hyperparameters of optimization, the structure in the gradient noise, and the Hessian matrix at the end of training that explains this anomalous diffusion. To build this understanding, we first derive a continuous-time model for SGD with finite learning rates and batch sizes as an underdamped Langevin equation. We study this equation in the setting of linear regression, where we can derive exact, analytic expressions for the phase-space dynamics of the parameters and their instantaneous velocities from initialization to stationarity. Using the Fokker-Planck equation, we show that the key ingredient driving these dynamics is not the original training loss but rather the combination of a modified loss, which implicitly regularizes the velocity, and probability currents that cause oscillations in phase space. We identify qualitative and quantitative predictions of this theory in the dynamics of a ResNet-18 model trained on ImageNet. Through the lens of statistical physics, we uncover a mechanistic origin for the anomalous limiting dynamics of deep neural networks trained with SGD. Understanding the limiting dynamics of SGD, and its dependence on various important hyperparameters like batch size, learning rate, and momentum, can serve as a basis for future work that can turn these insights into algorithmic gains.

The computational role of the abundant feedback connections in the ventral visual stream is unclear, enabling humans and nonhuman primates to effortlessly recognize objects across a multitude of viewing conditions. Prior studies have augmented feedforward convolutional neural networks (CNNs) with recurrent connections to study their role in visual processing; however, often these recurrent networks are optimized directly on neural data or the comparative metrics used are undefined for standard feedforward networks that lack these connections. In this work, we develop task-optimized convolutional recurrent (ConvRNN) network models that more correctly mimic the timing and gross neuroanatomy of the ventral pathway. Properly chosen intermediate-depth ConvRNN circuit architectures, which incorporate mechanisms of feedforward bypassing and recurrent gating, can achieve high performance on a core recognition task, comparable to that of much deeper feedforward networks. We then develop methods that allow us to compare both CNNs and ConvRNNs to finely grained measurements of primate categorization behavior and neural response trajectories across thousands of stimuli. We find that high-performing ConvRNNs provide a better match to these data than feedforward networks of any depth, predicting the precise timings at which each stimulus is behaviorally decoded from neural activation patterns. Moreover, these ConvRNN circuits consistently produce quantitatively accurate predictions of neural dynamics from V4 and IT across the entire stimulus presentation. In fact, we find that the highest-performing ConvRNNs, which best match neural and behavioral data, also achieve a strong Pareto trade-off between task performance and overall network size. Taken together, our results suggest the functional purpose of recurrence in the ventral pathway is to fit a high-performing network in cortex, attaining computational power through temporal rather than spatial complexity.

The neural plausibility of backpropagation has long been disputed, primarily for its use of non-local weight transport — the biologically dubious requirement that one neuron instantaneously measure the synaptic weights of another. Until recently, attempts to create local learning rules that avoid weight transport have typically failed in the large-scale learning scenarios where backpropagation shines, e.g. ImageNet categorization with deep convolutional networks. Here, we investigate a recently proposed local learning rule that yields competitive performance with backpropagation and find that it is highly sensitive to metaparameter choices, requiring laborious tuning that does not transfer across network architecture. Our analysis indicates the underlying mathematical reason for this instability, allowing us to identify a more robust local learning rule that better transfers without metaparameter tuning. Nonetheless, we find a performance and stability gap between this local rule and backpropagation that widens with increasing model depth. We then investigate several non-local learning rules that relax the need for instantaneous weight transport into a more biologically-plausible "weight estimation" process, showing that these rules match state-of-the-art performance on deep networks and operate effectively in the presence of noisy updates. Taken together, our results suggest two routes towards the discovery of neural implementations for credit assignment without weight symmetry: further improvement of local rules so that they perform consistently across architectures and the identification of biological implementations for non-local learning mechanisms.